A possible solution towards the above question is quantum kernels [1]}. As shown in Figure REF , there is a close correspondence between classical kernels and quantum kernels: the feature map \(\phi (\cdot )\)  coincides with the preparation of quantum states via variational quantum circuits \(U_E(\mathbf {x}^{(i)})\in \mathbb {C}^{2^N \times 2^N}\)  [2]}, [3]}, [4]}, i.e., \(\mathinner {|{\varphi (\mathbf {x}^{(i)})}\rangle }=U_E(\mathbf {x}^{(i)})\mathinner {|{\mathbf {0}}\rangle }\) , which map the input data into high-dimensional Hilbert spaces described by \(N\)  qubits but can not be effectively accessible; the result of kernel function \(\kappa (\cdot , \cdot )\)  coincides with applying measurements on the prepared quantum states, i.e., \(|\mathinner {\langle {\varphi (\mathbf {x}^{(j)})|\varphi (\mathbf {x}^{(i)})}\rangle }|^2\) , which enables the efficient collection of information from feature space. Due to the flexibility of the variational quantum circuits, quantum kernels have been experimentally implemented on different platforms such as superconducting, optical, and nuclear magnetic resonance quantum chips to resolve classification tasks [5]}, [6]}, [7]}, [8]}. As indicated by [9]}, quantum kernels can achieve advantages when the prepared quantum states are classically intractable. Following the same routine, Huang et al. [10]} recently proved the predication advantages of quantum kernels. Namely, for appropriate datasets, quantum kernels assures a lower generalization error bound than that of classical kernels [11]}.
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